The inverse of the star discrepancy of a union of randomly shifted Korobov rank-1 lattice point sets depends polynomially on the dimension

TL;DR

This work tackles the inverse star discrepancy problem by transferring a recent probabilistic construction from polynomial lattices to classical Korobov rank-1 lattices modulo a prime. Using Fourier analysis on $\mathbb{Z}_N$ and a four-way classification of generator selection (random vs. fixed) and shift type (continuous vs. discrete), the authors prove high-probability discrepancy bounds for multiset unions of shifted lattices: $D_{N_{tot}}^*(P) \le C\cdot\frac{ s\log(N+1) + \log 2 - \log(1-\delta) }{M}$ with $C \approx 1.83$ or $1.73$, depending on the regime. These bounds yield an $O\left(\frac{ s\log N_{tot} }{\sqrt{N_{tot}}} \right)$ decay and imply a quadratic dependence of $N(\varepsilon, s)$ on the dimension, matching analogous results for polynomial lattices while significantly narrowing the search to a finite set of lattice parameters and shifts. The results extend a probabilistic constructive path toward explicit, high-dimensional QMC point sets with near-optimal dimension scaling.

Abstract

The inverse of the star discrepancy, $N(ε, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most $ε$ in dimension $s$, is known to depend linearly on $s$. However, explicit constructions achieving this optimal linear dependence remain elusive. Recently, Dick and Pillichshammer (2025) made significant progress by showing that a multiset union of randomly digitally shifted Korobov polynomial lattice point sets almost achieve the optimal dimension dependence with high probability. In this paper, we investigate the analog of this result in the setting of classical integer arithmetic using Fourier analysis. We analyze point sets constructed as multiset unions of Korobov rank-1 lattice point sets modulo a prime $N$. We provide a comprehensive analysis covering four distinct construction scenarios, combining either random or fixed integer generators with either continuous torus shifts or discrete grid shifts. We prove that in all four cases, the star discrepancy is bounded by a term of order $O(s \log(N_{tot}) / \sqrt{N_{tot}})$ with high probability, where $N_{tot}$ is the total number of points. This implies that the inverse of the star discrepancy for these structured sets depends quadratically on the dimension $s$. While the proofs are probabilistic, our results significantly reduce the search space for optimal point sets from a continuum to a finite set of candidates parameterized by integer generators and random shifts.

Theorems & Definitions (31)

• Lemma 3.1
• proof
• Lemma 3.2
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• Lemma 3.3
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• Lemma 3.4
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• Lemma 3.5
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